Which of the following statements with respect to $f(x)$ and $g(x)$ are FALSE?
Solution 1:
Let $$ \int_0^1 tf(t)dt=A, \int_0^1t^2f(t)dt= B. $$ Then $$ f(x)=(1+A)x+Bx^2. $$ Note that $$ A=\int_0^1xf(x)dx=\int_0^1(1+A)x^2dx+\int_0^1Bx^3dx=\frac{1}{3}(1+A)+\frac14B. \tag1$$ Similarly, $$ B=\frac{1}{4}(1+A)+\frac15B. \tag2 $$ Solving $A,B$ from (1) and (2), you will get $f(x)$. I omit the detail.